Question

show that \[\huge e^{sinx}-e^{-sinx}-4=0\] has no real solution

Answer 1

say \(\large t = e^{\sin x}\)

Answer 2

take \[\large e^{sinx}=y\]

Answer 3

you will get a quadratic

Answer 4

so u'll get a equation like this
\[y-(1/y)-4=0\rightarrow y^2-4y-1=0\]

Answer 5

now solve it

Answer 6

im getting
y=2+sqrt(5)
and y=2-sqrt(5)

Answer 7

yes take
y=2+sqrt(5)

Answer 8

so
\[\large e^{sinx}=2+\sqrt{5}\]

Answer 9

\[sinx=\log_{e} (2+\sqrt{5})\]

Answer 10

again
\[2+\sqrt{5}>e \rightarrow \log_{e} (2+\sqrt{5})>\log_{e}e>1\]

Answer 11

hence sinx>1
which is impossible
so no real solution:)

Answer 12

ahh thank u :D